2025 SSMO Relay Round 4 Problems/Problem 3
Problem
Let 
A particle moves in the coordinate plane such that at any time 
its position is ![\[\left(\sum_{a=1}^{T-1} \cos(at),\sum_{a=1}^{T-1} \sin(at)\right).\]](http://latex.artofproblemsolving.com/2/a/f/2aff036f84c6f9ae0cd1f2684885b7c4fc9f0a5d.png)
Over the time interval ![$t\in(0,k],$](http://latex.artofproblemsolving.com/1/d/6/1d6e4f5955235095c51ab884b0552eb11476aea9.png)
the particle lies on at least one coordinate axes 
times. If the minimal value of 
can be written as 
for relatively prime positive integers 
and 
find 
.
